Autonomous routing algorithms, such as BGP, are intended to reach a globally consistent set of routes after nodes iteratively and independently collect, process, and share network information. Generally, the important role of the mechanism used to share information has been overlooked in previous analyses of these algorithms. In this paper, we explicitly study how the network-communication model affects algorithm convergence. To do this, we consider a variety of factors, including channel reliability, how much information is processed from channels, and how many channels are processed simultaneously. Using these factors, we define a taxonomy of communication models and identify particular models of interest, including those used in previous theoretical work, those that most closely model real-world implementations of BGP, and those of potential interest for the design of future routing algorithms. We characterize an extensive set of relationships among models in our taxonomy and show that convergence depends on the communication model in nontrivial ways. These results highlight that certain models are best for proving conditions that guarantee convergence, while other models are best for characterizing conditions that might permit nonconvergence.